1970 IMO Problems/Problem 1
Let be a point on the side of . Let , and be the inscribed circles of triangles , and . Let , and be the radii of the exscribed circles of the same triangles that lie in the angle . Prove that
We use the conventional triangle notations.
Let be the incenter of , and let be its excenter to side . We observe that
Simplifying the quotient of these expressions, we obtain the result
Thus we wish to prove that
But this follows from the fact that the angles and are supplementary.
By similar triangles and the fact that both centers lie on the angle bisector of , we have , where is the semi-perimeter of . Let have sides , and let . After simple computations, we see that the condition, whose equivalent form is is also equivalent to Stewart's Theorem Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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